I had the same problem recently. I’m no expert but this is what I did.
If you’re using Win 7 SP1 Home Premium:
click on the little ‘up’ arrow in your sys tray > click Customize > you’ll see 2 columns: Icons and Behaviors > find Avast! > set the Behavior for it as you want (show icon and notifications, hide icon and notifications, only show notifications). You may need to Restart after this.

If Avast! is not found in the Customize section, and you have the desktop icon shortcut on your desktop: Right click on it and select “Pin to Taskbar” Won’t be in the sys tray, but will be to the left of it on the bottom of your screen in the task bar. Not sure if it will show Avast! behavior (spin) but at least you can easily find it.

Hope this helps!